On solving singular interface problems using the enriched partition‐of‐unity finite element methods

Abstract

It has been well recognized that interface problems often contain strong singularities which make conventional numerical approaches such as uniform h‐ or p‐version of finite element methods (FEMs) inefficient. In this paper, the partition‐of‐unity finite element method (PUFEM) is applied to obtain solution for interface problems with severe singularities. In the present approach, asymptotical expansions of the analytical solutions near the interface singularities are employed to enhance the accuracy of the solution. Three different enrichment schemes for interface problems are presented, and their performances are studied. Compared to other numerical approaches such as h‐p version of FEM, the main advantages of the present method include: easy and simple formulation; highly flexible enrichment configurations; no special treatment needed for numerical integration and boundary conditions; and highly effective in terms of computational efficiency. Numerical examples are included to illustrate the robustness and performance of the three schemes in conjunction with uniform h‐ or p‐refinements. It shows that the present PUFEM formulations can significantly improve the accuracy of solution. Very often, improved convergence rate is obtained through enrichment in conjunction with p‐refinement.